Abstract

This paper introduces the theme of killing‐them‐softly between set‐theoretic universes. The main theorems show how to force to reduce the large cardinal strength of a cardinal to a specified desired degree. The killing‐them‐softly theme is about both forcing and the gradations in large cardinal strength. Thus, I also develop meta‐ordinal extensions of the hyper‐inaccessible and hyper‐Mahlo degrees. This paper extends the work of Mahlo to create new large cardinals and also follows the larger theme of exploring interactions between large cardinals and forcing central to modern set theory.